Notes on Refraction of Light at Plane Surfaces and Refractive Index

Refraction of Light at Plane Surfaces: Fundamental Concepts

• Definition of Reflection: The return of light into the same medium after striking a surface.

• Laws of Reflection:

  1. The angle of reflection is equal to the angle of incidence.

  2. The incident ray, the normal at the point of incidence, and the reflected ray all lie in the same plane.

• Definition of Refraction: The change in the direction of the path of light when it passes from one transparent medium to another. This is essentially a surface phenomenon.

• The Medium and Speed relationship: Light travels at different speeds in different media. The speed is maximum in a vacuum and remains constant within a transparent homogeneous medium.

• Refractive Scenarios and Deviation ($\delta$):

  • Rarer to Denser Medium: When light travels from a rarer medium (like air) to a denser medium (like glass), it slows down and bends towards the normal. In this case, \angle r < \angle i, and the deviation is defined as $\delta = i - r$.

  • Denser to Rarer Medium: When light travels from a denser medium to a rarer medium, it speeds up and bends away from the normal. In this case, \angle r > \angle i, and the deviation is defined as $\delta = r - i$.

  • Normal Incidence: If a ray is incident normally ($i = 0^{\circ}$), it passes undeviated ($r = 0^{\circ}$). The speed changes, but the direction does not. The deviation is $\delta = 0^{\circ}$.

Optical Density vs. Physical Density

• Optical Density: Dependent on the speed of light within the medium. A medium where light slows down is "optically denser" compared to the previous medium. A medium where light speeds up is "optically rarer."

• Physical Density: Refers to the mass per unit volume, typically dependent on inter-molecular separation.

• Key Distinction: Optical density has no direct relation to physical density.

  • Example: Kerosene floats on water, meaning its physical density is lower than water. However, kerosene is optically denser than water because light travels slower in kerosene than in water.

The Laws of Refraction (Snell’s Laws)

Refraction follows two specific laws, named after the Dutch scientist Willebrod Snell:

1. First Law: The incident ray, the refracted ray, and the normal at the point of incidence all lie in the same plane.

2. Second Law: The ratio of the sine of the angle of incidence ($i$) to the sine of the angle of refraction ($r$) is constant for a pair of given media.

   • Mathematical Expression: $\frac{\sin(i)}{\sin(r)} = \text{constant} = {_1\mu_2}$

   • The constant is known as the refractive index of the second medium with respect to the first medium ($\mu$).

Refractive Index (\mu) and Speed of Light

• Absolute Refractive Index: Defined as the ratio of the speed of light in vacuum (or air) to the speed of light in that specific medium.

  • Formula: $\mu = \frac{\text{Speed of light in vacuum (c)}}{\text{Speed of light in medium (V)}}$

  • Since V < c in any medium, $\mu$ is always greater than $1$.

• Units: Refractive index is a ratio of two similar quantities (speeds) and thus has no unit.

• Relative Refractive Index: Refractive index of medium 2 with respect to medium 1 is given by:

  • ${_1\mu_2} = \frac{V_1}{V_2} = \frac{\mu_2}{\mu_1}$

• Standard Speed Values:

  • Speed of light in air/vacuum ($c$): $3 \times 10^8\,\text{m s}^{-1}$ (Precise value: $299,792,458\,\text{m s}^{-1}$).

  • Speed of light in water: $2.25 \times 10^8\,\text{m s}^{-1}$.

  • Speed of light in glass: $2 \times 10^8\,\text{m s}^{-1}$.

• Calculated Examples:

  • Refractive index of glass ($\mu_{\text{glass}}$): $\frac{3 \times 10^8}{2 \times 10^8} = 1.5$.

  • Refractive index of water ($\mu_{\text{water}}$): $\frac{3 \times 10^8}{2.25 \times 10^8} = \frac{4}{3} = 1.33$.

  • Refractive index of diamond: $2.41$ (signifies light travels $2.41$ times faster in air than in diamond).

Refractive Index Table of Common Substances

Effect of Refraction on Speed, Wavelength, and Frequency

• Frequency ($f$): Does not change during refraction. It is determined by the source of light.

• Speed ($V$): Decreases when moving from rarer to denser medium; increases when moving from denser to rarer medium.

• Wavelength ($\lambda$): Changes in proportion to speed to keep frequency constant.

  • Equation: $V = f \times \lambda$

  • When passing from air (speed $c$, wavelength $\lambda$) to a medium (speed $V$, wavelength $\lambda'$):

    • $f = \frac{c}{\lambda} = \frac{V}{\lambda'}$

    • $\lambda' = \frac{V}{c} \times \lambda$

    • Given $\mu = \frac{c}{V}$, then $\lambda' = \frac{\lambda}{\mu}$

• Transitions:

  • Rarer to Denser (\mu > 1): Wavelength decreases (\lambda' < \lambda).

  • Denser to Rarer (\mu < 1): Wavelength increases (\lambda' > \lambda).

Factors Affecting Refractive Index

1. Nature of the Medium (Optical Density): The higher the optical density (slower speed of light), the higher the refractive index.

2. Physical Conditions (Temperature): As temperature increases, the speed of light in the medium increases, causing the refractive index to decrease.

3. Color or Wavelength of Light:

   • Speed of all colors is identical in vacuum but different in transparent media.

   • Speed of Red light is maximum; speed of Violet light is least.

   • Refractive index is maximum for violet and least for red (\mu_V > \mu_R).

   • Refractive index decreases with an increase in wavelength ($\mu \propto \frac{1}{\lambda}$).

Principle of Reversibility and Undeviated Rays

• Principle of Reversibility: The path of a light ray is reversible. If a ray travels from Medium 1 to 2, a ray incident from Medium 2 back alongside the path of the refracted ray will emerge along the path of the original incident ray.

  • Relationship: ${_1\mu_2} \times {_2\mu_1} = 1$ or ${_1\mu_2} = \frac{1}{{_2\mu_1}}$.

  • Example: Refractive index of glass wrt air is $\frac{3}{2}$, then air wrt glass is $\frac{2}{3}$.

  • Relative Index Calculation: ${_w\mu_g} = \frac{\mu_g}{\mu_w} = \frac{1.5}{1.33} = \frac{9}{8} = 1.125$.

• Conditions for Undeviated Rays:

  1. The angle of incidence is zero ($i = 0^{\circ}$).

  2. The refractive indices of both media are identical ($\mu_1 = \mu_2$).

Experimental Verification and Determination of Refractive Index

• Apparatus: Rectangular glass block, white sheet, drawing board, pins.

• Procedure:

  1. Fix the paper and trace the block boundary ($PQRS$).

  2. Draw a normal ($NOM$) and an incident path ($AO$) at an angle $i$ (e.g., $40^{\circ}$).

  3. Fix two pins ($a$ and $b$) on $AO$.

  4. Looking from the opposite side ($RS$), fix pins $c$ and $d$ such that all four pins appear to be in a single straight line.

  5. Trace the refracted ray ($OB$) and emergent ray ($BC$).

• Determination:

  • Measure $i$ and $r$, then find $\frac{\sin(i)}{\sin(r)}$.

  • Alternative Method: Draw a circle from center $O$, intersecting the incident ray at $D$ and refracted ray at $E$. Drop normals $DF$ and $EG$ to the main normal. The ratio $\mu = \frac{DF}{EG}$ since radii $OD$ and $OE$ are equal.

Refraction through a Rectangular Glass Block

• Geometry: Refraction occurs at two parallel surfaces ($PQ$ and $RS$).

• Angle of Emergence: By the principle of reversibility, the angle of incidence ($i$) is equal to the angle of emergence ($e$).

• Directionality: The emergent ray is parallel to the incident ray but is shifted laterally.

• Lateral Displacement ($x$): The perpendicular distance between the path of the emergent ray and the direction of the original incident ray.

  • Formula: $x = \frac{t \times \sin(i-r)}{\cos(r)}$ (where $t$ is the thickness of the block).

• Factors affecting Lateral Displacement:

  1. Thickness of medium: Displacement increases with thickness.

  2. Angle of incidence: Displacement increases with angle $i$.

  3. Refractive index: Displacement increases with higher refractive index.

  4. Wavelength: Displacement is more for violet light than for red light because \mu_V > \mu_R.

Multiple Images in a Thick Glass Plate or Mirror

• When viewing an object obliquely through a thick glass mirror (silvered on the back surface), multiple images are seen.

• Formation Process:

  1. First Surface Reflection: A small portion (about $4\%$) of light reflects off the top surface, forming a faint virtual image ($A_1$).

  2. Primary Refraction: The majority (about $96\%$) of light refracts into the glass.

  3. Strong Back Reflection: Light reflects off the silvered back surface strongly.

  4. Second Virtual Image ($A_2$): This ray refracts back into the air. $A_2$ is the brightest image because it results from the first strong reflection at the silvered surface.

  5. Subsequent Reflections: Internal light continues to reflect and refract multiple times, creating images $A_3, A_4, A_5, \dots$ of gradually decreasing brightness.